## Bonus-malus systems with two-component mixture models arising from different parametric families.(English)Zbl 1393.62048

Summary: Two-component mixture distributions defined so that the component distributions do not necessarily arise from the same parametric family are employed for the construction of Optimal Bonus-Malus Systems (BMSs) with frequency and severity components. The proposed modeling framework is used for the first time in actuarial literature research and includes an abundance of alternative model choices to be considered by insurance companies when deciding on their Bonus-Malus pricing strategies. Furthermore, we advance one step further by assuming that all the parameters and mixing probabilities of the two component mixture distributions are modeled in terms of covariates. Applying Bayes’ theorem we derive optimal BMSs either by updating the posterior probability of the policyholders’ classes of risk or by updating the posterior mean and the posterior variance. The resulting tailor-made premiums are calculated via the expected value and variance principles and are compared to those based only on the a posteriori criteria. The use of the variance principle in a Bonus-Malus ratemaking scheme in a way that takes into consideration both the number and the costs of claims based on both the a priori and the a posterior classification criteria has not yet been proposed and can alter the resulting premiums significantly, providing the actuary with useful alternative tariff structures.

### MSC:

 62P05 Applications of statistics to actuarial sciences and financial mathematics 91B30 Risk theory, insurance (MSC2010)

GAMLSS
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### References:

 [1] Boucher, J. P.; Denuit, M.; Guillen, M., Risk classification for claim counts: A comparative analysis of various zero-inflated mixed Poisson and hurdle models, North American Actuarial Journal, 11, 4, 110-131, (2007) [2] Boucher, J. P.; Denuit, M.; Guillen, M., Models of insurance claim counts with time dependence based on generalisation of Poisson and negative binomial distributions, Variance, 2, 1, 135-162, (2008) [3] Brouhns, N.; Guillen, M.; Denuit, M.; Pinquet, J., Bonus-malus scales in segmented tariffs with stochastic migration between segments, Journal of Risk and Insurance, 70, 577-599, (2003) [4] Denuit, M.; Marechal, X.; Pitrebois, S.; Walhin, J. F., Actuarial Modelling of Claim Counts: Risk Classification, Credibility and Bonus-Malus Systems, (2007), Wiley · Zbl 1168.91001 [5] Dionne, G.; Vanasse, C., A generalization of actuarial automobile insurance rating models: the negative binomial distribution with a regression component, ASTIN Bulletin, 19, 199-212, (1989) [6] Dionne, G.; Vanasse, C., Automobile insurance ratemaking in the presence of asymmetrical information, Journal of Applied Econometrics, 7, 149-165, (1992) [7] Dunn, P. K.; Smyth, G. K., Randomized quantile residuals, Computational and Graphical Statistics, 236-245, (1996) [8] Frangos, N.; Vrontos, S., Design of optimal bonus-malus systems with a frequency and a severity component on an individual basis in automobile insurance, ASTIN Bulletin, 31, 1, 1-22, (2001) · Zbl 1035.62108 [9] Gómez, E.; Hernández, A.; Vázquez-Polo, F., Robust Bayesian premium principles in actuarial science, Journal of the Royal Statistical Society, 49, 241-252, (2000) [10] Gómez, E.; Pérez, J.; Hernández, A.; Vázquez-Polo, F., Measuring sensitivity in a bonus–malus system, Insurance: Mathematics & Economics, 31, 105-113, (2002) · Zbl 1037.62110 [11] Gómez-Déniz, E.; Hernández-Bastida, A.; Fernández-Sánchez, M. P., Computing credibility bonus-malus premiums using the aggregate claims distribution, Hacettepe Journal of Mathematics and Statistics, 43, 6, 1047-1061, (2014) · Zbl 1367.91086 [12] Heilmann, W., Decision theoretic foundations of credibility theory, Insurance: Mathematics & Economics, 8, 77-95, (1989) · Zbl 0687.62087 [13] Heller, G. Z.; Stasinopoulos, M. D.; Rigby, R. A.; de Jong, P., Mean and dispersion modeling for policy claims costs, Scandinavian Actuarial Journal, 4, 281-292, (2007) · Zbl 1164.91030 [14] Johnson, N. L.; Kotz, S.; Kemp, A. W., Univariate Discrete Distributions, (2005), Wiley [15] Klein, N.; Denuit, M.; Lang, S.; Thomas, K., Nonlife ratemaking and risk management with Bayesian generalized additive models for location, scale, and shape, Insurance: Mathematics and Economics, 55, 225-249, (2014) · Zbl 1296.62089 [16] Lemaire, J., Bonus-Malus Systems in Automobile Insurance, (1995), Kluwer Academic Publishers [17] Picech, L., The merit-rating factor in a multiplicating rate-making model, ASTIN Colloquium, (1994) [18] Pinquet, J., Allowance for cost of claims in bonus-malus systems, ASTIN Bulletin, 27, 33-57, (1997) [19] Pinquet, J., Designing optimal bonus-malus systems from different types of claims, ASTIN Bulletin, 28, 205-220, (1998) · Zbl 1162.91430 [20] Rigby, R. A.; Stasinopoulos, D. M., Generalized additive models for location, scale and shape, (with discussion), Applied Statistics, 54, 507-554, (2005) · Zbl 05188697 [21] Rigby, R. A.; Stasinopoulos, D. M., A flexible regression approach using GAMLSS in R, (2009) [22] Stasinopoulos, D. M.; Rigby, B.; Akantziliotou, C., instructions on how to use the GAMLSS package in R, (2008) [23] Tzougas, G.; Frangos, N., The design of an optimal bonus-malus system based on the sichel distribution, Modern Problems in Insurance Mathematics, 239-260, (2014) · Zbl 1327.91039 [24] Tzougas, G.; Vrontos, S.; Frangos, N., Optimal bonus-malus systems using finite mixture models, ASTIN Bulletin, 44, 2, 417-444, (2014) · Zbl 1288.91120
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